The optimal sensor placement problem for the estimation of the temperature distribution in buildings is a highly nonlinear and multi-scale problem where stochastic perturbations are usually present. The main goal here is to properly locate sensors in order to reliably estimate the temperature distribution in certain areas. Since feedback controllers are usually in use, a proper estimation of the state is of utmost importance in order to reduce energy consumption of such controllers. Closely related to the sensor placement problem is the design of certain features in buildings that also would improve the estimation of the state. This issue includes location of inflow and outflow vents and of heat sources and insulated regions.

Optimal Sensor Placement

The main objective consists in locating of a finite number of sensors within an admissible location while a certain minimum distance among them is maintained and a prescribed criterium for information quality is optimized.

Although ideally the sensor/actuator location and design problems are dealt with simultaneously, we focus primarily on sensor location since it is cheaper to re-locate sensor networks (than to re-design and/or re-locate actuators). Furthermore, in most scenarios, the control system is a feedback closed-loop controller that benefits from the better quality of the information coming from the re-located sensor.

In mathematical terms, we assume $\Omega$ is a room or building where sensors are placed and there is a ventilation system involving outlets and inlets; see Figure 1. The temperature of the room on the time interval $(0,T)$ is determined by $u$ such that $u:(0,T)\times\Omega\to\mathbb{R}$, and it is advected by the velocity profile of the air $\mathbf{v}:(0,T)\times \Omega\to \mathbb{R}^n$ . In general, we assume that the velocity profile satisfies the Navier-Stokes equation
\begin{equation}\tag{NS}\label{eq:NS}
\begin{aligned}
\partial_t \mathbf{v}+\mathbf{v}\cdot \nabla \mathbf{v}-(1/\mathrm{Re})\Delta\mathbf{v}+\nabla p&=0, \qquad \text{ on } (0,T)\times\Omega
\end{aligned}
\end{equation}
with additional compressibility assumptions, mixed boundary conditions and $\mathbf{v}(0, \cdot)=\mathbf{v}_0$ on $\Omega$. Here, $\mathrm{Re}$ is the Reynolds number, $p:(0,T)\times \Omega\to \mathbb{R}$ denotes the pressure, and $\mathbf{v}_0:\Omega\to \mathbb{R}^n$ is prescribed. In Figure 2, we find the streamlines obtained from $\mathbf{v}$ from the example of Figure 1.

The convection-diffusion process is perturbed by an stochastic (Wiener) process $\eta$. We measure $u$ with a sensor in location $x$ by a weighted average within an effective range and obtain an output $h_x$ which is also perturbed by an stochastic process $\nu$. The system satisfied by these variables is given by
\begin{align*}
\partial_tu&=\alpha\Delta u+\mathbf{v}\cdot\nabla u+ \eta;\\
h_x&=C(x)u+\nu;
\end{align*}
where the sensor action is determined by $ C(x)w=\int_{\Omega}K(y-x)w(y) dy,$ and with $K$ some kernel function defined by properties of the sensor. An appropriate criteria for where sensor is placed is obtained by trying to minimize the expected value of $\|u-\tilde{u}_x\|^2_{L^2(\Omega)}$ where $\tilde{u}_x$ is the output of the Kalman Filter .
Even for simple problems, the isosurfaces of
\begin{equation}
x\mapsto J(x):=\int_0^1 \mathbb{E}\|u-\tilde{u}_x\|_{L^2(\Omega)}^2 dt
\end{equation}
seem to be associated to non-convex problems. This is evidenced (for the example of Figure 1) by Figure 3 and 4, where the distribution of $x\mapsto J(x)$ on the walls, and the associated isosurfaces for $x\mapsto J(x)$ inside the domain are shown, respectively.

Upon discretization, for the computation of $J(x)$, the problem reduces (in many cases) to the resolution of a sequence of large-scale algebraic Riccati equations: $A\Sigma+\Sigma A^*+BB^*-\Sigma C^* C\Sigma=0$ where $A\in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m}$, $C\in \mathbb{R}^{p\times n}$, and $n$ is significantly large: within the range $10^4-10^6$.
This implies that several computational and storage challenges are present, since even
the storage of $A$ can easily exceed memory capacity.

Figure 3.- Colormap of $\partial\Omega\ni x\mapsto J(x)$.

Figure 4.- Isosurfaces of $\Omega\ni x\mapsto J(x)$.

Robust Sensor Placement

An important problem in sensor placement concerns the robustness of the sensor location with respect to perturbations $\mathbf{h}$ of the velocity profile $\mathbf{v}$. This pertains to the issue of the sensitivity of the map $\mathbf{h}\mapsto J(x;\mathbf{h})$ defined as
\begin{equation}
J(x;\mathbf{h}):=\int_0^1 \mathbb{E}\|u^{\mathbf{h}}-\tilde{u}_x^{\mathbf{h}}\|_{L^2(\Omega)}^2 dt,
\end{equation}
where $u^{\mathbf{h}}$ is the solution to the convection-difussion stochastic process with velocity profile $\mathbf{v}+\mathbf{h}$ and $\tilde{u}_x^{\mathbf{h}}$ the output of the associated generalized Kalman-Bucy filter.
In fact, a sensor location may be optimal for a specific velocity profile, but adding a small perturbation to it may render this location subpar. This is indeed the situation encountered within the placement of thermostats in large buildings.
A number of obstacles and questions, in the theoretical and numerical aspects, are found in this setting. They involve appropriate function space frameworks for the study of sensitivities, study of the stochastic properties of $u$ and tailored model reduction techniques for the discretization and further numerical implementation of the problem.

Once identified the space of admissible perturbations $\mathbf{H}(\Omega)$ and provided that $\mathbf{h}\mapsto J(x;\mathbf{h})$ is directional differentiable, then an appropriate measure of robustness for sensor location is
\begin{equation}
G(x):=\sup_{|\mathbf{z}|_{\mathbf{H}(\Omega)}\leq M} D_{\mathbf{h}}J(x;\mathbf{0})\mathbf{z},
\end{equation}
where $D_{\mathbf{h}}J(x;\mathbf{0})\mathbf{z}$ is the directional derivative at zero in the direction $\mathbf{z}$, and $M>0$ is an energy bound of the perturbations. It follows that lower values of $G$ are desirable.
The accurate estimation functional $J$ and the robustness measure $G$ are now two objective functionals that we desire to minimize simultaneously. However, their structure is significantly different: good places of estimation are not robust places, and vice-versa.
In Figure 5, we observe the stationary velocity profile $\mathbf{v}$ solution to Navier-Stokes in a geometry resembling an airport. In Figure 6, $x\mapsto J(x)$ is depicted, and the location of an optimal place for estimation is observed. Figure 7 shows the structure of the robustness functional $x\mapsto G(x)$ when perturbations are generated on the inlet closest to the least robust place. It becomes apparent that both structures are non-trivial are highly geometry dependent.

Figure 5.- Velocity profile $\mathbf{v}$ solution to Navier-Stokes in a prototypical airport with many inlets/outlets.

Figure 6.- Accurate estimation functional $x\mapsto J(x)$. Note that corners are edges are bad estimation locations.

Figure 7.-Robustness measure functional $x\mapsto G(x)$. Note that the scale depends on the perturbation bound $M>0$.

Publications related to the project

Refereed publications and preprints of submitted articles

M. Hintermüller, C. N. Rautenberg, and S. Rösel. Density of convex intersections and applications Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 473(2205), 2017.

M. Hintermüller, M. Kanitsar, and C. N. Rautenberg. Shape optimization for Navier-Stokes flow with mixed boundary conditions and industrial applications.

[2]

M. Hintermüller, K. Papafitsoros, and C. N. Rautenberg . Variable step mollifiers and applications.

[3]

C. N. Rautenberg, and W. Hu. Optimal Feedback Controllers and Sensor Placement for the Boussinesq System.

[4]

A. Alphonse, M. Hintermüller, and C. N. Rautenberg. Stability of the solution set and optimal control of quasi-variational inequalities.

Project Related Talks

[T1]

On the optimal sensor placement problem at the XXIV Congreso de Ecuaciones Diferenciales y Aplicaciones / XIV Congreso de Matemática Aplicada (XXIV CEDYA / XIV CMA), University of Cadiz (Universidad de Cádiz), Cádiz, Spain, June 8-12, 2015.

[T2]

Robust Optimal Sensor Placement at the IV International Conference on Applied Mathematics, Design and Control: Mathematical Methods and Modelling in Engineering and Life Sciences, San Martin National University (Universidad Nacional de San Martín), Buenos Aires, Argentina, November 4-6, 2015.

[T3]

Optimal and Robust Sensor Placement at Computational Methods for Control of Infinite-dimensional Systems, University of Minnesota, Minneapolis (MN), US, March 14-18, 2016.

[T4]

Optimal Sensor Location in Distributed Parameter Systems at International Workshop Optimal Control of Dynamical Systems, Graz, Austria, May 19, 2017.

[T5]

Stability and control of quasi-variational inequalities with non-unique solutions at 14th International Conference on Free Boundary Problems: Theory and Applications, Shanghai Jiao Tong University, Shanghai China, US, July 9-14, 2017.

[T6]

Robust Sensor Placement for Quasi-Variational Inequalities at Sensor Location in Distributed Parameter Systems, University of Minnesota, Minneapolis MN, US, September 06-08, 2017.

[T7]

Optimization problems with quasi-variational inequality constraints at the Oberwolfach Workshop ``Challenges in Optimal Control of Nonlinear PDE-Systems'', Oberwolfach, Germany, April 8-14, 2018.

[T8]

On the optimal control of quasi-variational inequalities at the 23rd International Symposium on Mathematical Programming (ISMP), Bordeaux, France, July 1-6, 2018.